Under what circumstances can preferences under uncertainty be modeled by expected utility? The classic von Neumann-Morgenstern theorem constructs a utility function that, when multiplied by corresponding probabilities, yields an ordering that is consistent with the ordering of the preferences of the agent.
Being able to represent choice under uncertainty via expectations is a powerful tool. As just one example, utility functions tend to be convex to capture diminishing marginal utility. Hence, by Jenson's inequality, the notion of risk aversion (the utility of a certain outcome of 1 dollar is higher than the utility of a 50% chance of 2 dollars and 50% chance of zero dollar) pops out of such models "for free". More generally, the rich mathematical and optimization frameworks that accompany expectations is made available for economic modeling.
The von Neumann-Morgenstern theorem itself relies on two core axioms of economics, a third axiom that is required to prove the existence of a utility function, and finally a fourth that is required to prove the existence of a utility function such that the expectation of the utility in each possible state corresponds to the actual preference ordering of the agent. It is this fourth axiom that is the lynchpin for the von Neumann-Morgenstern proof.
The late Daniel Kahneman, Nobel prize winner and critic of "mainstream" economics, took issue with this fourth axiom. (As an aside, his pop-science book "Thinking Fast and Slow" took (unjustified) issue with far more in economics than just this fourth axiom. He repeatedly conflates the concept of rationality and of the representation of preferences via expectations, leading the casual reader to believe that he had essentially undermined all of economics). As a simple example, the "Allais Paradox" posits two gambles that would appear to contradict the fourth axiom. In response to this shortcoming, he created an entirely new framework he termed prospect theory.
So does Kahneman's work prove the death-knell for expected utility? I don't believe so. There is a trick for resolving Allais' Paradox while retaining the expected utility framework: switching the "real-world" probabilities into an equivalent measure. A simple re-weighting of the probabilities generates expected utility that is consistent with the actual preference ordering, and it is not difficult to construct a general re-weighting that accomplishes this for any "gamble" in which at least one state has a higher utility than the alternative. Put in such a way, the fourth axiom is actually not necessary for expectations to be an appropriate representation of preferences. By adding the single line "under some probability measure" to the von Neumann-Morgenstern theorem, the entire analytical toolkit of expected utility is back at our disposal.
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I think some of this is also that you didn't really give a thorough explanation, but just an overview. At least, that's what's going on in my case. I'd otherwise be pretty interested.
The Allais' case would be where E[u(X)]<E[u(Y)] even though X is preferred to Y. But as long as there is a non-zero probability that the outcome of X (let's call it x) is preferred to the outcome of Y (y), the probabilities can be re-weighted to put more emphasis on the state of the world in which x>y. A trivial example would be simply giving weight only to the probabilities for which x>y.
In practice, in a discrete probability space, the re-weighted probabilities can be represented as a non-negative least-squares problem. Let U be a matrix which holds the utility of each gamble in each state, multiplied by the real-world probability of each state. Let b be the actual utility of the gamble. Then minimize | | Ax-b | |. The altered probabilities xp/sum(xp) then form the measure v under which E_v[u(X)]>E_v[u(Y)].
Okay, so what does solve the allais paradox mean in this case? What's the benefit of this? Am I right in reading you as saying that it continues to be the case that agents violate the axioms of expected utility in the Allais case, but that you can still use those preferences and transform them into a non-violating utility function that uses the agent's preferences to make it, but is not actually representative of the agent?
No, the Allais paradox is only a violation of the 4th axiom from the von Neummann-Morgenstern theorem. This axiom is necessary to show the existence of a representation of preferences by an expectation under the "real world" measure. My point is that the 4th axiom isn't necessary for such a representation. The measure under which expectations must be taken may need to change, but the use of expectations is still valid. Its not the utility function but the probabilities that are changed.
I don't get this. Why can you freely change the probabilities? Don't they represent something about the real world?
They can yes, and there may be cases (I can't think of one off the top of my head) where the actual probabilities are required. But if the objective is to be able to represent preferences via expectations of utility functions (and take advantage of all the benefits that such a representation can provide), then such a representation can be achieved under an equivalent measure instead. Its not too different conceptually from using "risk neutral" probabilities in asset pricing.
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